Simplify the rational expression. (2x^(2)-5x-12)/(x^(2)-7x+12)

Factor Numerator and Denominator: Factor the numerator and the denominator. The numerator 2x^2 - 5x - 12 and the denominator x^2 - 7x + 12 can be factored into products of binomials. For the numerator, we look for two numbers that multiply to 2*(-12) = -24 and add up to -5. These numbers are -8 and 3. For the denominator, we look for two numbers that multiply to 1*12 = 12 and add up to -7. These numbers are -3 and -4.Factor Using Found Numbers: Factor the numerator and the denominator using the numbers found in Step 1. The numerator factors to (2x - 8)(x + 3). The denominator factors to (x - 3)(x - 4).Re-evaluate Factoring: Simplify the rational expression by canceling out common factors. We notice that (x + 3) is not a common factor, but there seems to be a mistake in the factoring process as there is no common factor to cancel out. We need to re-evaluate the factoring of the numerator.

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Q. Simplify the rational expression.

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\frac{2 x^{2}-5 x-12}{x^{2}-7 x+12}

Factor Numerator and Denominator: Factor the numerator and the denominator. $\newline$ The numerator $2x^2 - 5x - 12$ and the denominator $x^2 - 7x + 12$ can be factored into products of binomials. $\newline$ For the numerator, we look for two numbers that multiply to $2*(-12) = -24$ and add up to $-5$ . These numbers are $-8$ and $3$ . $\newline$ For the denominator, we look for two numbers that multiply to $1*12 = 12$ and add up to $-7$ . These numbers are $-3$ and $-4$ .
Factor Using Found Numbers: Factor the numerator and the denominator using the numbers found in Step $1$ . $\newline$ The numerator factors to $(2x - 8)(x + 3)$ . $\newline$ The denominator factors to $(x - 3)(x - 4)$ .
Re-evaluate Factoring: Simplify the rational expression by canceling out common factors. $\newline$ We notice that $(x + 3)$ is not a common factor, but there seems to be a mistake in the factoring process as there is no common factor to cancel out. We need to re-evaluate the factoring of the numerator.